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In this paper we propose a generalized Markov Graph model for social networks and evaluate its application in social network synthesis, and in social network classification. The model reveals that the degree distribution, the clustering coefficient distribution as well as a newly discovered feature, a crowding coefficient distribution, are fundamental to characterizing a social network. The application of this model to social network synthesis leads to a capacity to generate networks dominated by the degree distribution and the clustering coefficient distribution. Another application is a new social network classification method based on comparing the statistics of their degree distributions and clustering coefficient distributions as well as their crowding coefficient distributions. In contrast to the widely held belief that a social network graph is solely defined by its degree distribution, the novelty of this paper consists in establishing the strong dependence of social networks on the degree distribution, the clustering coefficient distribution and the crowding coefficient distribution, and in demonstrating that they form minimal information to classify social networks as well as to design a new social network synthesis tool. We provide numerous experiments with published data and demonstrate very good performance on both counts.